Volume Distortion in Groups

Suppose H is a finitely presented group, and let w be a word representing the identity in H. Then w can be written as the product of conjugates of relators; the minimal such number is defined to be the area of w. If H is a subgroup of G, also finitely presented, then it is possible that the area of w in G is much less than its area in H. We will define the area distortion function, which provides a measure for this difference, and then generalize this to a volume distortion function. We will provide bounds in terms of the Dehn functions of the groups and find the volume distortion for abelian subgroups of abelian-by-cyclic groups.